Jaycee Dugard

Few people could understand what the three women rescued this week from apparent captivity in Cleveland must have gone through better than Jaycee Dugard.

Abducted herself in 1991 at age 11 and held captive for 18 years, Dugard is cautioning the public to give these women time and space to begin healing. At the same time, she adds, the women's survival shows the power of human resolve.

"These individuals need the opportunity to heal and connect back into the world. This isn't who they are. It is only what happened to them," Dugard tells PEOPLE in a statement.

These days, Jaycee Dugard rides horses. Working with the animals is therapeutic, even confidence-boosting, she says.

"You have to be very sure of yourself when you're riding," she tells ABC News's Diane Sawyer in an interview airing Tuesday (6:30 p.m. ET). "You don't want to have any doubts in your mind because they'll sense that."

It's a motto that seems to have a deeper meaning for Dugard, who was kidnapped in 1991 when she was 11 years old and held captive for 18 years by Phillip and Nancy Garrido.

"My name is Jaycee Dugard, and I want to say that because for a long time I wasn't able to say my name and so it feels good," the soft-spoken woman told the crowd at an awards ceremony Friday night in New York. "I am truly honored to be here tonight with these amazing women who have done and been through so much more than me."

Dugard, 31, who had been kidnapped at age 11 and held prison for 18 years, appeared at her first public event since her ordeal.

Though parole agents were supposed to be supervising Garrido at his home in Antioch, Calif., from 1991 to 1999, they never detected Dugard's presence there during that time, the complaint states. Dugard, 31, who gave birth to two of Garrido's children during the years she was held captive, was not freed until 2009.

Several disturbing home videos of young children frolicking in public, made by Jaycee Dugard's abductors two decades ago, have been released by the El Dorado County district attorney's office as part of an attempt to prevent similar tragedies from happening in the future.

Two clips feature young children playing in public in the late '80s or early '90s, California's Mountain Democrat reports.

One shows Phillip Garrido playing a guitar and singing in park, although the camera is focused on the kids playing in the background.

"Everything she says makes you stop and examine yourself and your life," the news anchor, whose exclusive interview with Dugard aired Sunday on a two-hour ABC Primetime special, told PEOPLE for the July 18 issue. "She will astonish you."

Asked if Dugard was emotional during the interview, Sawyer, 65, said she saw "lots of different moods, and when she goes back in time, you can see it in her eyes, you can see it in her face: She remembers with such vivid clarity what it was like to be 11, in this nightmare … What you see moving through her eyes is unforgettable." – Caroline Slutsky

With her abductors pleading guilty and sentenced to life in prison, the details are finally coming out about the fear and pain that the then-11-year-old Jaycee Dugard endured during her kidnapping.

Dugard, now 31, described her ordeal in testimony before a grand jury last September, and on Thursday the court released a heavily redacted transcript. But even in the shortened form, the transcript's account – the first to be made public – is chilling.

Jaycee Dugard broke her silence about her 18 years in captivity, lashing out at her abductors in a statement read in court Thursday by her mother.

"I chose not to be here today because I refuse to waste another second of my life in your presence," she said in the remarks read at the sentencing for Phillip and Nancy Garrido. "Everything you ever did was wrong and I hope one day you will see that."

"I hated every day for 18 years," the statement continues. "You stole my life and that of my family."

As she read, Dugard's mother Terry Probyn became increasingly upset. Her voice got louder until she finally shouted, "I hate you … for the torment you put my family through!"